This note deals with an invariant of colored links. Let us detail
these objects.

An oriented link with $n$ components is a set of $n$ disjoint smooth
oriented closed curves embedded in $S^{3}$. An oriented link is colored if each component is provided with a color. In other words,
a link is colored if a function $\gamma$ is defined on the set of
components with values in a finite set $N$ of colors. Let’s call
such a function $\gamma$ a coloration. Every coloration
introduces an equivalence relation in the set of components: two
components are equivalent if they have the same color. Let $C$ be
the set of components of a link $L$. Two colorations $\gamma$ and
$\gamma^{\prime}$ of $L$ are said to be equivalent if there is a
bijection in $N$ between the images $\gamma(C)$ and $\gamma^{\prime}(C)$,
which takes, for every $c\in C$, the color $\gamma(c)$ into the
color $\gamma^{\prime}(c)$. Two colorations define the same partition of the
set $C$ into classes of equivalence if and only if they are
equivalent.

An invariant of colored links takes the same value on isotopic links
with the same coloration, and may take different values on the same
link with different colorations. Our invariant of colored links
takes the same value on isotopic links with equivalent colorations.

There is a wide literature on the invariants of colored links
developed from the multivariable Alexander polynomial, which
includes contributions by Fox, Conway, Kauffman, and many others.
For a summary of the history of these invariants see [Cimasoni2004].
More recent works in this direction ([Murakami1992]; [Cimasoni2004]; [Cimasoni and Florens2008]) use also skein relations for the Alexander polynomial, but the
colors of the components of the links involved in such skein
relations are unaltered.

The skein relation introduced in the present note may change the
colors of the components of the link. Therefore, the invariant that
we define seems to be new; still, the relation of it with the known
invariants has to be investigated.

We recall here the definition of an invariant of classical links by
a skein relation.

Among the most famous topological invariants of oriented knots and
links, we single out the Alexander polynomial, introduced in
[Alexander1928], and the Jones polynomial, introduced in [Jones1987].
Both these polynomials in one variable can be considered as
particular instances of a more general invariant of isotopy classes
of oriented links, the so-called HOMFLY polynomial, introduced in
[Freyd et al.1985], which is a Laurent polynomial in two variables. This
polynomial is uniquely defined by the condition that its value is 1
on the unknotted oriented circle and by a skein relation, as we
explain below.

Suppose that $K_{\mathsf{x}}^{+}$, $K_{\mathsf{x}}^{-}$ and $K_{\mathsf{x}}^{0}$ are diagrams of three
oriented links that are exactly the same except in the neighborhood
of a crossing $\mathsf{x}$, where they look as shown in the figure below.
Then the skein relation is an equation that relates the values of
the polynomial $P$ on these links:

The proof that the skein relation allows to define the value of $P$
on any link is done by induction on the number of crossings. By an
ordered sequence of crossing changes, any link $K$ can be changed
into a trivial link $K_{u}$, i.e. a collection of unknotted and
unlinked components. The value of $P$ on $K_{u}$ is supposed known:
indeed, it must be independent of the number $n$ of crossings,
depending only on the number of components. Let $(\mathsf{x}_{1},\ldots,\mathsf{x}_{m})$
be the sequence of crossings, where the sign has to be changed to
transform $K$ into $K_{u}$, and let ${}_{i}K$ be the diagram obtained from
$K$ by changing the sign of the first $i$ crossings, so that
${}_{m}K=K_{u}$. The skein relation is thus applied to the crossing
$\mathsf{x}_{1}$: if this crossing is positive in $K$, then $K=K_{\mathsf{x}_{1}}^{+}$,
and one obtains $P(K_{\mathsf{x}_{1}}^{+})$ as a linear combination of the
values $P(K_{\mathsf{x}_{1}}^{-})$ and $P(K_{\mathsf{x}_{1}}^{0})$. Since $K_{\mathsf{x}_{1}}^{0}$ has
$n-1$ crossings, $P(K_{\mathsf{x}_{1}}^{0})$ is known by the induction
hypothesis. Now, by definition, $K_{\mathsf{x}_{1}}^{-}=\ _{1}K$, and the value
$P(_{1}K)$ is calculated by using the skein relation applied to the
second crossing $\mathsf{x}_{2}$, and so on, until the last crossing change,
in which either ${}_{(m-1)}K_{\mathsf{x}_{m}}^{+}$ or ${}_{(m-1)}K_{\mathsf{x}_{m}}^{-}$
coincides with $K_{u}$. Then the calculation of $P(K)$ is concluded,
being obtained from $P(K_{u})$ and the values of $P$ on $(m-1)$ links
with $n-1$ crossings.

It is proved that $\mathfrak{P(}K)$ calculated in this way is independent of
the chosen sequence of changes, as well as independent of the
particular projection of the link. The polynomial $P$ is in fact an
isotopy invariant of the link.

Suppose now that we want to define by a skein relation an invariant,
which is able to distinguish two links that are identical, but have
non-equivalent colorations. We show that it is possible to obtain
the invariant we want by introducing a new skein relation, which
takes into account strands of any color, and by setting the value of
the invariant on the trivial links with non-equivalent colorations,
i.e., on collections of $n$ unknotted and unlinked components that
are colored, for every $c=1,\ldots,n$, with $c$ different colors.

Our invariant, $F$, requires three variables, say $x$, $w$, and $t$.
When the link is colored with a sole color, our invariant is reduced
to another instance of the HOMFLY polynomial and becomes the Jones
polynomial in the variable $t$ if $w=t^{1/2}$.

Finally, we remark that the polynomial $F$ was not found by
searching for an invariant for colored links. Its definition is a
byproduct of the study of another class of links, the tied
links (see [Aicardi and Juyumaya2015]), obtained from the closure of tied braids,
which, in turn, constitute a diagrammatic representation of an
abstract algebra, the so-called algebra of braids and ties.
The defining set of generators of the algebra of braids and ties
consists of two sets of generators, one of them is interpreted as
the set of usual braid generators and the other one as the set of
ties. Despite the fact that the defining relations, which involve
tie generators, look complicated and, in particular, ties do not
commute with braids, we have shown ([Aicardi and Juyumaya2014]) that a new
geometric interpretation of the tie generators, coherent with the
defining relations of the algebra, allows in particular to reduce
ties to simple connections between different strands of braids, and,
under closure, between different components of links. The set of
connections behaves as an equivalence relation among the link
components. As a consequence, the invariant polynomial for tied
links defined in [Aicardi and Juyumaya2015] provides an invariant of colored links
as well.

A colored link diagram looks like a diagram of a link, where each
component is colored.

Definition 1.

Two oriented colored links are c-isotopic if they are ambient
isotopic and their colorations are equivalent (Fig. 1).

Figure 1: The first two diagrams represent c-isotopic colored
links, the third one is not c-isotopic to the others. Black and gray
represent two different colors

Figure 2: a The discs, where $\mathsf{K}_{+}$ and $\mathsf{K}_{-}$ are not
c-isotopic. b The discs, where $\mathsf{K}_{\sim}$, $\mathsf{K}_{+,\sim}$, and
$\mathsf{K}_{-,\sim}$ are not c-isotopic. Black and gray indicate any two
colors

Let $R$ be a commutative ring, and $\mathcal{C}$ be the set of colored
oriented link diagrams. By an invariant of colored links, we
mean a function $I:\mathcal{C}\rightarrow R$, which is constant on each
class of c-isotopic links.

Remark 1.

In the sequel, we denote by $\mathsf{K}$ an oriented colored link as well as
its diagram, if there is no risk of confusion.

Theorem 1.

There exists a rational function in the variables $x,w,t$, $F:\mathcal{C}\rightarrow\mathbb{Q}(x,t,w)$, invariant of oriented colored links,
uniquely defined by the following three conditions on colored-link
diagrams:

I

The value of $F$ is equal to 1 on the unknotted circle.

II

Let $\mathsf{K}$ be a colored link with $n$ components and $c$
colors. By $[\mathsf{K};\mathsf{O}]$, we denote the colored link with $n+1$
components consisting of $\mathsf{K}$ and the unknotted and unlinked circle
colored with the $(c+1)$-st color. Then

$\displaystyle F([\mathsf{K};\mathsf{O}])=\frac{1}{wx}F(\mathsf{K}).$

III

(skein relation) Let $\mathsf{K}_{+}$ and $\mathsf{K}_{-}$ be the diagrams of
two colored links that are the same outside a small disc into which
two strands enter, and inside this disc look as shown in Fig. 2a. Black and gray indicate any two colors, not
necessarily distinct. Similarly, let $\mathsf{K}_{\sim}$ and $\mathsf{K}_{+,\sim}$ be
two links that inside the disc look as shown in Fig. 2b, while outside the disc coincide with $\mathsf{K}_{+}$ and
$\mathsf{K}_{-}$, except, possibly, for the colors of all the components of
the link having the same colors as the strands entering the disc;
all these components in $\mathsf{K}_{\sim}$ and $\mathsf{K}_{+,\sim}$ are colored by a
sole color. Then the following identity holds:

Remark 2.

Skein relation III holds for any two colors of the strands
involved. In particular, if the two colors coincide, then
$\mathsf{K}_{+}=\mathsf{K}_{+,\sim}$ and $\mathsf{K}_{-}=\mathsf{K}_{-,\sim}$, so that relation III is
reduced to relation

If $n>c$, then there are two circles colored by the same color.
Regard them as $\mathsf{K}_{{}_{\displaystyle\widetilde{\ \ }}}$. Then
$\mathsf{K}_{+}=\mathsf{K}_{-}=\mathsf{O}_{n-1}^{c}$ (see figure).

Proof of Theorem1 Theorem 1 is proved by
the procedure used in the proof of the corresponding theorem for
classical links stated on page 112 in [Lickorish and Millet1987]. We will outline
the parts, where the presence of colors modifies the demonstration.

Of course, the Reidemeister moves for colored links are all the
Reidemeister moves, where the strands involved may have different
colors.

One starts with zero crossings as the induction base: the value of
$F$ on the unlink $O_{n}^{c}$ with $n$ circles and $c$ colors is given
by $\displaystyle\frac{y^{n-c}}{(wx)^{n-1}}$ in accordance with
Remark 3.

Let $\mathcal{C}^{p}$ be the set of diagrams of oriented colored links with $p$
crossings, and $\mathsf{K}\in\mathcal{C}^{p}$. By ordering the components and fixing a
point in each component, one constructs, for every diagram $\mathsf{K}$, an
associated standard ascending diagram $\mathsf{K}^{\prime}$ in the following way:
traverse the components of $\mathsf{K}$ in their given order and from their
base points in the direction specified by their orientation. Every
crossing, encountered the first time, is either over or under
crossing. In the first case, the crossing is changed; otherwise, it
is left as in $\mathsf{K}$. $\mathsf{K}^{\prime}$ consists of the same number of components
as $\mathsf{K}$, completely unknotted and unlinked. $\mathsf{K}$ and $\mathsf{K}^{\prime}$ are
identical, except for a finite sequence of crossings, which we call
’deciding’, where the signs are opposite. Furthermore, we define
${\tilde{\mathsf{K}}}^{\prime}$ as obtained from $\mathsf{K}^{\prime}$ by coloring with a sole color
all the components that have the same two colors as the two strands
involved into each deciding crossing. $\mathsf{K}^{\prime}$ and
${\tilde{\mathsf{K}}}^{\prime}$ are, by construction, collections of unknotted and
unlinked components; $\mathsf{K}^{\prime}$ has the same number of colors as $\mathsf{K}$,
${\tilde{\mathsf{K}}}^{\prime}$ may have a smaller number of colors. The procedure
defining $\mathsf{K}^{\prime}$ allows to get an ordered sequence of deciding
crossings, whose order depends on the ordering of the components and
on the choice of the base points.

The induction hypothesis states that, on $\mathcal{C}^{p}$, we have a well
defined function $F$, satisfying relations I–III, which is
independent of the ordering of the components, independent of the
choices of the base points, and invariant under Reidemeister moves
that do not increase the number of crossings beyond $p$. Moreover,
the induction hypothesis states that the value of $F$ on any colored
link with $p$ crossings, consisting of $n$ components unknotted,
unlinked, and colored with $c$ colors ($c\leq n$), is the same as
$F(\mathsf{O}_{n}^{c})$ given by (3).

Otherwise, construct the associate ascending diagrams $\mathsf{K}^{\prime}$ and
${\tilde{\mathsf{K}}}^{\prime}$ and consider the first deciding crossing $\mathsf{x}$. If,
in a neighborhood of $\mathsf{x}$, the colored link looks like $\mathsf{K}_{+,\sim}$
(or $\mathsf{K}_{-,\sim}$), use skein relation IV to write the value of $F$
in terms of $\mathsf{K}_{-,\sim}$ and $\mathsf{K}_{\sim}$ (respectively,
$\mathsf{K}_{+,\sim}$ and $\mathsf{K}_{\sim}$). If, in a neighborhood of $\mathsf{x}$, the
colored link looks like $\mathsf{K}_{+}$ (respectively, $\mathsf{K}_{-}$), use skein
relation Va (respectively, Vb) to write the value of $F$ in terms of
the value of $F$ on the colored links $\mathsf{K}_{-},\mathsf{K}_{-,\sim}$, and
$\mathsf{K}_{\sim}$ (respectively, $\mathsf{K}_{+},\mathsf{K}_{+,\sim}$, and $\mathsf{K}_{\sim}$).
Observe that if the diagram $\mathsf{K}_{\epsilon}$ or $\mathsf{K}_{\epsilon,\sim}$
($\epsilon=\pm$) coincides, in a neighborhood of the crossing $\mathsf{x}$,
with the original diagram, then $\mathsf{K}_{-\epsilon}$ or
$\mathsf{K}_{-\epsilon,\sim}$ coincides, in the neighborhood of the same
crossing, with the associated diagram $\mathsf{K}^{\prime}$ or ${\tilde{\mathsf{K}}}^{\prime}$
respectively. On the other hand, $\mathsf{K}_{\sim}$ represents a colored
link diagram with $p$ crossings, for which the value of $F$ is known
and is invariant according to the induction hypothesis. Then we look
for the second deciding crossing, which is present in all the
diagrams with $p+1$ crossings, obtained by applying the skein
relation to $\mathsf{K}$ at $\mathsf{x}$. We apply the same procedure to such
diagrams at the second deciding crossing, and so on. The procedure
ends with the last deciding crossing, thus yielding diagrams of
unlinked colored links with $p+1$ crossings, where the function is
given by (4), which depends only on the number of components
and the number of colors. For an example see Sect. 1.2.

It remains to prove that:

(1)

the procedure is independent of the order of the deciding
points;

(2)

the procedure is independent of the order of components
and of the choice of base-points;

(3)

the polynomial $F$ is invariant under Reidemeister moves.

Following the proof done in [Lickorish and Millet1987] for classical links, we
observe that the proofs of statements (1), (2), and (3) can be done
in an analogous way also in presence of colors. Of course, every
time a skein relation is used, we have to pay attention to the
colors of all links involved. We give here the proof of statement
(1) as an example. The proofs of the other statements are similar.

The proof of statement (1) consists in a verification that the value
of the invariant does not change if we interchange any two deciding
crossings in the procedure of calculation. So, let $\mathsf{K}$ be the
diagram of a colored link, and let $\mathsf{x}$ and $\mathsf{y}$ the first two
deciding crossings that will be interchanged.

Denote by $\epsilon_{\mathsf{x}}$ the sign at the crossing $\mathsf{x}$, by $\sigma_{\mathsf{x}}\mathsf{K}$ the colored link diagram obtained from $\mathsf{K}$ by
changing the sign at the crossing $\mathsf{x}$, by $\tilde{\sigma}_{\mathsf{x}}\mathsf{K}$ the
diagram obtained from $\mathsf{K}$ by changing the sign at $\mathsf{x}$ and coloring
by a sole color all the components colored by the two colors of the
two components crossing at $\mathsf{x}$; then denote by $\rho_{\mathsf{x}}\mathsf{K}$ the
diagram obtained from $\mathsf{K}$ by removing the crossing $\mathsf{x}$ and
coloring by a sole color all the components colored by the two
colors of the components crossing at $\mathsf{x}$. Then do the same for the
crossing $\mathsf{y}$.

If $\mathsf{x}$ follows $\mathsf{y}$, then $F(\mathsf{K})$ is obtained from the above
expression by interchanging $\mathsf{x}$ with $\mathsf{y}$. Observe that this
expression contains terms of type $\tau_{\mathsf{y}}\tau_{\mathsf{x}}\mathsf{K}$ or
$\alpha(\tau_{\mathsf{x}}\tau^{\prime}_{\mathsf{y}}\mathsf{K}+\tau^{\prime}_{%
\mathsf{x}}\tau_{\mathsf{y}}\mathsf{K})$, where
$(\tau,\tau^{\prime})\in\{\sigma,\tilde{\sigma},\rho\}$ and $\alpha$ is
a coefficient. Such terms are invariant under the interchange of
$\mathsf{x}$ with $\mathsf{y}$, because the operation $\tau_{\mathsf{x}}$ commutes with
$\tau_{\mathsf{y}}$ as well as with $\tau^{\prime}_{\mathsf{y}}$ (since the procedures of
coloring by a sole color the components of the two colors of the
strands crossings respectively at $\mathsf{x}$ and at $\mathsf{y}$ produce
equivalent colorations of the link if they are interchanged.
Therefore, $F(\mathsf{K})$ is independent of the order of $(\mathsf{x},\mathsf{y})$.
$\square$

Here we list some properties of the polynomial $F$, which can be
easily verified.

(i)

$F$ is multiplicative with respect to the connected sum
of colored links.

(ii)

The value of $F$ does not change if the orientations of
all curves of the link are reversed.

(iii)

Let $\mathsf{K}$ be a link diagram whose components are all
tied together, and $\mathsf{K}^{\pm}$ be the link diagram obtained from $\mathsf{K}$
by changing the signs of all crossings. Then $F(\mathsf{K}^{\pm})$ is obtained
from $F(\mathsf{K})$ by the following changes: $w\rightarrow 1/w$ and $t\rightarrow 1/t$.

(iv)

Let $\mathsf{K}$ be a knot or a link, whose components have all
the same color. Then $F(\mathsf{K})$ is defined by relations I and IV, and
is a HOMFLY polynomial with

Statement (i) is deduced from defining relation I of $F$, by the
same arguments that prove the multiplicativity of the invariants
obtained by skein relations (see [Lickorish and Millet1987]). Statement (ii) is
evident, since the value of $F$ on the unlinked circles is
independent of their orientation, and the skein relations are
invariant under the inversion of the orientations of the strands.
Statement (iii) follows from the fact that, if $w$ and $t$ are
replaced by $1/w$ and $1/t$ respectively, then the skein relations
Va and Vb are interchanged, whereas the term $y/(xw)$ stays
unchanged.

To prove statement (iv), we compare skein relation IV, multiplied by
$t^{1/2}$, with Eq. (1); this provides the
expressions of $\ell$ and $m$ in terms of $x,y$, and $t$.
Furthermore, since for the Jones polynomial $V(\mathsf{K})$

Observe also that $\mathsf{K}_{3}$ is c-isotopic to the oriented trefoil with
three negative crossings, and that $\mathsf{K}_{1}$ is c-isotopic to two
oriented circles of different colors linked by two negative
crossings.